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On some properties of free commutators with semicircular variables

Mihai Popa, Kamil Szpojankowski

TL;DR

The paper analyzes free commutators with a semicircular variable $s$ and a free $x$, establishing that the distribution of $s+i[s,x]$ equals the free additive convolution $mu_s oxplus mu_{i[s,x]}$ despite $s$ and $i[s,x]$ not being freely independent. It provides a precise combinatorial expression for the free cumulants of $x+i[x,s]$, and proves that $x+i[x,s]$ is freely infinitely divisible whenever $x$ is FI ext{D}, aided by an operator-model construction that realizes these cumulants as moments of a positive measure. The work extends previous results to general $x$ and connects cumulant formulas with a constructive framework, including a Fock-type operator model, to reveal how FI ext{D} properties are preserved under commutators in free probability. These results deepen understanding of free additive convolution in nonfree settings and offer explicit tools for studying infinite divisibility in free probability.

Abstract

We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution identity \[ μ_{s + i[x, s]} = μ_s \boxplus μ_{i[x, s]}. \] Furthermore, we prove that the polynomial \( x + i[x, s] \) is freely infinitely divisible whenever \( x \) itself is freely infinitely divisible.

On some properties of free commutators with semicircular variables

TL;DR

The paper analyzes free commutators with a semicircular variable and a free , establishing that the distribution of equals the free additive convolution despite and not being freely independent. It provides a precise combinatorial expression for the free cumulants of , and proves that is freely infinitely divisible whenever is FI ext{D}, aided by an operator-model construction that realizes these cumulants as moments of a positive measure. The work extends previous results to general and connects cumulant formulas with a constructive framework, including a Fock-type operator model, to reveal how FI ext{D} properties are preserved under commutators in free probability. These results deepen understanding of free additive convolution in nonfree settings and offer explicit tools for studying infinite divisibility in free probability.

Abstract

We investigate commutators of free variables of the form , where is a semicircular element. We show that although and are not free, their sum nevertheless satisfies the free additive convolution identity \[ μ_{s + i[x, s]} = μ_s \boxplus μ_{i[x, s]}. \] Furthermore, we prove that the polynomial is freely infinitely divisible whenever itself is freely infinitely divisible.

Paper Structure

This paper contains 5 sections, 9 theorems, 53 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free convolution without freeness $s+i[s,x]$
  4. Free cumulants of $x+i[x,s]$
  5. Operator model

Key Result

Theorem 1.1

Let $s$ and $x$ be free random variables such that $s$ is semicircular. Then the distribution of $s + i[s, x]$ is the free additive convolution of the distributions of $s$ and $i[s, x]$. Moreover, the elements $s$ and $i[s, x]$ are not free.

Figures (8)

  • Figure 1: Example of partition $\sigma\in \mathrm(8)$ with 4 blocks together with $\pi\in NC(4)$ and corresponding partition $\pi(\sigma)\in NC(8)$
  • Figure 2: Configuration in which singleton $s$ on position $i$ is paired with $s$ on position $j$ coming from a pair $xs$.
  • Figure 3: Configuration in which singleton $s$ on position $i$ is paired with $s$ on position $j$ coming from a pair $sx$ with $i<j$.
  • Figure 4: Graphical representation of the bijection $\Phi$.
  • Figure 5: Graphical representation of bijection $\Phi$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Proposition 4.1
  • ...and 3 more